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QUALITY AND RELIABILITY CORNER
Optimal inspection frequency
Optimal
inspection
frequency
A tool for maintenance planning/forecasting
Subhash Mathew
763
Department of Mechanical Engineering, Monash University, Melbourne,
Australia
Keywords Maintenance programmes, Failure (mechanical), Maintenance costs
Abstract Maintenance management is expected to plan for all maintenance activities for the
life of the equipment. It must be able to forecast and plan the future maintenance requirements
of spares, man-hours and total costs. As equipment ages and enters the wear-out stage, with
increasing failure rates, this forecasting becomes difficult. Maintenance management is faced
with the dilemma of either resorting to high inventories, over-planning and inflated budgets or
of suffering stockouts, lengthy delays in repair and budget blowouts. A model for an optimal
inspection frequency can help correct this. For an inspection frequency to be optimal, it must
exactly match the failure rate of the equipment. Hence, with the use of a cost rate factor, the
optimal inspection frequency can also be used as a tool for planning and forecasting
maintenance costs. This paper develops an optimal model, ensuring that the inspection
frequency is capable of matching the varying failure rates throughout the life of the equipment.
It also demonstrates how this optimal inspection frequency can then be used to plan and
forecast maintenance costs.
1. Introduction
Maintenance management is expected to plan for all maintenance activities for the life
of the equipment. It must be able to forecast and plan the future maintenance
requirements of spares, man-hours and total costs (Sarker and Haque, 2000). As
equipment ages and enters the wear-out stage with increasing failure rates, this
forecasting becomes difficult. Maintenance management is faced with the dilemma of
either resorting to high inventories, over-planning and inflated budgets or of suffering
stockouts, lengthy delays in repair and budget blowouts. A model for an optimal
inspection frequency can help correct this. The development of an optimal model for an
inspection maintenance strategy and its use as a tool for planning and forecasting
maintenance costs are presented in this paper.
If a piece of equipment had a constant failure rate throughout its life, the optimal
inspection frequency and the maintenance costs over a unit of time would be constant.
However, over its lifetime, equipment can be expected to demonstrate decreasing,
constant and increasing rates of failure. Hence, in an optimal model, the inspection
frequency should be capable of matching these varying failure rates. Moreover, the
planning or forecasting of maintenance costs should also reflect this variation over
time. Hence, a direct relationship between optimal inspection frequency and
maintenance costs can be expected to give a good approximation of maintenance
costs. Thus an optimal model for inspection frequency not only minimizes overall costs
consisting of both breakdown and inspection costs, it also provides a powerful tool for
planning and forecasting maintenance requirements and costs.
International Journal of Quality &
Reliability Management
Vol. 21 No. 7, 2004
pp. 763-771
q Emerald Group Publishing Limited
0265-671X
DOI 10.1108/02656710410549109
IJQRM
21,7
764
Frequent inspection of equipment can help determine its true status and appropriate
actions can be taken to prevent failure. Hence, as affirmed by Ramakumar (1993), an
inspection maintenance strategy can reduce the downtime for equipment subject to
random failure. Bloch and Geitner (1983) also recommend an inspection maintenance
strategy, noting that almost all failure of equipment is preceded by indications of early
deterioration. An appropriate inspection maintenance strategy can detect these
indications and help prevent a breakdown failure (Ito and Nakagawa, 2000; Baohe,
2002; Rouhan and Schoefs, 2003; Fleming, 2004). However, every inspection also incurs
costs. Hence the goal is to create an optimal inspection maintenance strategy that
minimizes overall costs consisting of both breakdown and inspection costs.
Three models were proposed by Murthy and Naikan (1996) for the three stages of
equipment life. The probability density function is used to determine the inspection
policy in the infant mortality stage of the equipment. Lead-time for spares and the
mean time between failure determine the inspection policy for the constant failure rate
stage. Finally, in the wear-out stage, the vibration level of the equipment determines
the inspection policy.
2. Model development
2.1. Model for hazard or failure rate
The goal is to develop a model for the frequency of inspection, which provides the best
fit over the life of equipment, that is assumed to be a single unit or simple system
subject to stochastic failure. The development of a model for the frequency of
inspection is based on the hazard rate assumed for the life of the equipment. The
hazard rate itself is dependent on the probability density function assumed for the
equipment.
Research on the development of a model for inspection frequency has generally
favored assuming a negative exponential distribution of failure for the equipment. The
negative exponential distribution has been recommended in the models developed by
(Jardine, 1979; Rao and Varaprasad, 1985; Kececioglu, 1991; Wild, 1995). The various
models are variations on the basic model proposed by Jardine (1979). His model defines
the hazard rate as follows:
hðtÞ ¼ lðnÞ
ð1Þ
¼ ðk=nÞ
ð2Þ
The hazard or failure rate is shown as being inversely related to the inspection
frequency n and directly related to the arrival rate of breakdowns k over unit time. The
variations in the other models lie in redefining the value of k.
The chief limitation in assuming this distribution is that the inspection frequency and
the failure or hazard rate for this distribution are independent of time. This implies that
over its entire life the equipment has a failure rate that is largely constant. In practice this
would not be the case. Particularly in the wear-out stage, the equipment can be expected
to have a steadily increasing rate of failure. An effective inspection maintenance strategy
needs to take this into account. A model needs to be developed which is time dependent.
The bath tub-shaped failure rate has generally been accepted as being representative of
the hazard rate for equipment over time (Pulcini, 2001; Xie et al., 2002). The hazard
rate or failure rate curve for equipment consists of three distinct areas, namely:
(1) the early life stage, or DFR stage where the failure rate is decreasing;
(2) the useful life stage, or the CFR stage where the failure rate is constant; and
(3) the wear out stage, or the IFR stage where the failure rate is increasing.
0-T1
Represents the time interval for the early life stage or DFR stage
765
T1-T2 Represents the time interval for the useful life stage or CFR stage
T2+
Represents the time interval for the wear out stage or IFR stage
Using a shape parameter, the Weibull distribution can be shown to fit the failure
characteristics of equipment at different stages of its life, by merely changing the value
of the shape parameter appropriately (Xie et al., 2002). For the Weibull distribution, (h)
is the scale parameter, also referred to as the characteristic life parameter. The shape
parameter is (b). By varying the scale and shape parameters, a large number of
distributions can be approximated:
b , 1 represents; DFR; ðdecreasing; failure rateÞ stage
b ¼ 1 represents CFR ðconstant failure rateÞ stage
b . 1 represents IFR ðincreasing failure rateÞ stage
ð3Þ
2.2. Model for inspection frequency
Having considered the Weibull distribution as appropriate, one can consider the
constraints to define a model for the frequency of inspection. The model should satisfy
the following conditions:
.
The frequency of inspection should be time dependent.
.
When plotted against time, the slope of the function should be negative in the
DFR stage, constant in the CFR stage, and positive in the IFR stage. This would
indicate that any change in the failure rate is reflected appropriately in the
frequency of inspection also.
.
There should be no discontinuity at the transition stages in time at T1, when
DFR changes to CFR and also at T2 when CFR changes to IFR.
The first two parameters can be satisfied by making (n) the frequency of inspection as
a function of time. A shape factor (u) can then be introduced, such that a negative value
for (u) yields the DFR condition, a zero value the CFR condition and a positive value
yields the IFR condition. To satisfy the third condition, one needs to introduce a new
variable TT, for transition time, as shown in the definition of terms, listed below. The
model developed for inspection frequency and the conditions it is subjected to and
maintenance cost rate are all shown in the list. The model for inspection frequency may
be defined as follows:
nðtÞ ¼ N :ðt=T T Þu . . . subject to the parameters defined in the list
Definition of terms and their constraints:
T1 =
Optimal
inspection
frequency
Time of transition from DFR stage to CFR stage.
ð4Þ
IJQRM
21,7
T2 =
Time of transition from CFR stage to IFR stage.
TT =
Transition time variable
ðMTBFÞ0 ¼ MTBF for the useful life stage i.e. CFR stage
766
N=
(MTBF)-1
0
CRF
Maintenance cost rate factor
MCR
Maintenance cost rate, i.e. maintenance cost per unit time.
For the DFR stage
0 , t # T 1 ; u , 0; T T ¼ T 1 ;
For the CFR stage
T 1 , t , T 2 ; u ¼ 0; T T ¼ ðMTBFÞ0 ;
For the IFR stage
T 2 # t; u . 0; T T ¼ T 2 ;
Maintenance cost rate (MCR) i.e. the maintenance cost per unit of time for the equipment
may be defined, using the Cost rate factor (CRF) as follows:
MCR ¼ nðtÞ:CRF
For the Weibull distribution
EðMTBFÞ ¼ ðMTBFÞ0 ¼ ðhÞ{Gð1 þ 1=bÞ}
For the Gamma function
and Gð1 þ xÞ ¼ x!
Gð1 þ xÞ ¼ xGðxÞ
ð5Þ
ð6Þ
2.3 Application for optimal inspection frequency and maintenance costs
This analysis assumes the following for the equipment:
.
The early life stage, or DFR stage where the failure rate is decreasing, is from 0 to
10 time units.
.
The useful life stage, or the CFR stage where the failure rate is constant is from
10 to 100 time units.
.
The wear-out stage, or the IFR stage where the failure rate is increasing is
beyond 100 time units.
The Weibull distribution scale factor ðhÞ ¼ 10:
For the CFR stage ðbÞ ¼ 1 and ðuÞ ¼ 0:
T 1 ¼ 10; T 2 ¼ 100:
Optimal
inspection
frequency
ðMTBFÞ0 ¼ ðhÞ{Gð1 þ 1=bÞ}
¼ ð10Þ{Gð1 þ 1=1Þ}
¼ 10 Time units
nðtÞ ¼ N ¼ ðMTBFÞ21
0 ¼ ð1=hÞ ¼ 0:1
767
CRF Maintenance cost rate factor ¼ $10; 000 per optimal inspection
MCR ¼ Maintenance cost rate:
For CFR stage; MCR ¼ CRF :nðtÞ
¼ ${10; 000: 0:1} per unit time
¼ $1; 000 per unit time:
The DFR stage
0 , t # T 1 ; u , 0; T T ¼ T 1 ; T 1 ¼ 10:
Plots for the values of Shape factor (u) where u ¼ (2 0.25), (2 0.50), (2 0.75), (2 1.0)
(see also Table I and Figure 1).
The CFR stage
T 1 , t , T 2 ; u ¼ 0; T T ¼ ðMTBFÞ0 ¼ 10 Time units
N ¼ ðMTBFÞ21
0 ¼ ð1=hÞ ¼ 0:1; nðtÞ ¼ N :
See also Table II and Figure 2.
The IFR stage
T 2 # t; u . 0; T T ¼ T 2 ; T 2 ¼ 100:
Time in
units
1
2
3
4
5
6
7
8
9
10
Inspection frequency n(t)
Shape
Shape
Shape
Shape
factor
factor
factor
factor
(21.00)
(2 0.75)
(20.50)
(2 0.25)
0.18
0.15
0.14
0.13
0.12
0.11
0.11
0.11
0.1
0.1
0.32
0.22
0.18
0.16
0.14
0.13
0.12
0.11
0.11
0.1
0.56
0.33
0.25
0.2
0.17
0.15
0.13
0.12
0.11
0.1
1
0.5
0.33
0.25
0.2
0.17
0.14
0.13
0.11
0.1
Shape
factor
(2 0.25)
($)
1,800
1,500
1,400
1,300
1,200
1,100
1,100
1,100
1,000
1,000
Maintenance cost rate
Shape
Shape
factor
factor
(20.75)
(20.50)
($)
($)
3,200
2,200
1,800
1,600
1,400
1,300
1,200
1,100
1,100
1,000
Note: For values of the shape factor (u) from (2 0.25) to (21) in DFR stage
5,600
3,300
2,500
2,000
1,700
1,500
1,300
1,200
1,100
1,000
Shape
factor
(2 1.00)
($)
10,000
5,000
3,300
2,500
2,000
1,700
1,400
1,300
1,100
1,000
Table I.
Inspection frequency n(t)
and maintenance cost
rate versus time
IJQRM
21,7
768
Figure 1.
Inspection frequency in
DFR stage
Time in units
Table II.
Inspection frequency n(t)
and maintenance cost
rate versus time
Figure 2.
Inspection frequency in
CFR stage
10
20
30
40
50
60
70
80
90
100
Inspection frequency n(t)
Shape factor
(0)
Maintenance cost rate
Shape factor
(0) ($)
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
Note: When the value of the Shape factor (u) is zero in CFR stage
Plots for the values of shape factor (u), where u ¼ (3.0), (4.0), (5.0), (6.0) (see also
Table III and Figure 3).
3. Conclusions
Planning and forecasting maintenance requirements and costs can be helped greatly
by having a model for optimal inspection frequency. The model should be such that it
can correspond well to the changes in the failure rate of equipment. Previous research
has often relied on the negative exponential failure distribution. In real life, the failure
rate of equipment can be expected to vary with time. Hence, a time dependent
inspection frequency would be the preferred model. Accordingly, several conditions
were specified and a model developed to try and meet these conditions. As confirmed in
the model analysis, this model for inspection frequency satisfies all the three conditions
prescribed:
(1) The inspection frequency n(t) is time dependent.
(2) The inspection frequency reflects the variations in the hazard rate through
appropriate values for the shape factor (u). In the CFR stage where (u) equals
zero, the inspection frequency n(t) has a constant value of N.
Time in units
100
110
120
130
140
150
Shape
factor
(3)
0.1
0.13
0.17
0.22
0.27
0.34
Inspection frequency n(t)
Shape
Shape
Shape
factor
factor
factor
(4)
(5)
(6)
0.1
0.15
0.21
0.29
0.38
0.51
0.1
0.16
0.25
0.37
0.54
0.76
0.1
0.18
0.3
0.48
0.75
1.14
Shape
factor
(3) ($)
1,000
1,300
1,700
2,200
2,700
3,400
Optimal
inspection
frequency
769
Maintenance cost rate
Shape
Shape
Shape
factor
factor
factor
(4) ($)
(5) ($)
(6) ($)
1,000
1,500
2,100
2,900
3,800
5,100
1,000
1,600
2,500
3,700
5,400
7,600
1,000
1,800
3,000
4,800
7,500
11,400
Table III.
Inspection frequency n(t)
and maintenance cost
rate versus time for
values of the shape factor
(u) from (3) to (6) in IFR
stage
Figure 3.
Inspection frequency in
IFR stage
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21,7
770
(3) There is no discontinuity at the transition stages T1 and T2. At the first
transition stage as t approaches T1, (t/TT) approaches 1. Hence n(t) approaches
N. Similarly, at the second transition stage as t approaches T2, (t/TT)
approaches 1. Hence in this case also n(t) approaches N.
As it has satisfied all three conditions, one can assume that this model is the preferred
model for the optimal inspection frequency. In the useful life stage or CFR stage, the
MTBF of the equipment and the maintenance costs in an interval of time provide the
initial data and basis of analysis. As shown in Figure 3, maintenance management can
create a large number of scenarios using different values of the shape factor (u). This
helps determine the plot that best fits the data of their particular equipment. As shown,
when the shape factor (u) equals (3), the maintenance cost rate grows from $1,000 to
$3,400 over 50 units of time. However when the shape factor (u) equals (6), the
maintenance cost rate climbs to $11,400 in the same interval. Similar projections can
guide maintenance management in determining the time at which maintenance costs
become unsustainable and equipment replacement ought to be considered. Thus an
optimal model for inspection frequency not only minimizes overall costs consisting of
both breakdown and inspection costs, it also provides a powerful tool for planning and
forecasting maintenance requirements and costs and even for decisions for forecasting
equipment replacement.
Further research into this, in a collaborative effort with the equipment
manufacturers is required. The goal should be that based on research and the
database of equipment supplied, the manufacturer should be able to supply a band of
Weibull parameters, failure rate curves and transition points within which the actual
figures can be expected to lie. Eventually these figures could be tabulated for industry
type, usage and equipment size. This would greatly ease the work of maintenance
management.
References
Baohe, S. (2002), “An optimal inspection and diagnosis policy for a multi-mode system”,
Reliability Engineering and System Safety, Vol. 76, pp. 181-8.
Bloch, H.P. and Geitner, F.K. (1983), Machinery Failure Analysis and Troubleshooting, Gulf
Publishing, Houston, TX.
Fleming, K.N. (2004), “Markov models for evaluating risk-informed in-service inspection
strategies for nuclear power plant piping systems”, Reliability Engineering & System
Safety, Vol. 83 No. 1, pp. 27-45.
Ito, K. and Nakagawa, T. (2000), “Optimum inspection policies for a storage system with
degradation at periodic tests”, Math. Comput. Modelling, Vol. 31, pp. 191-5.
Jardine, A.K.S. (1979), “Solving industrial replacement problems”, Proceedings of Annual
Reliability and Maintainability Symposium, IEEE, pp. 136-41.
Kececioglu, D. (1991), Reliability Engineering Handbook, 2nd ed., Prentice-Hall, Englewood Cliffs,
NJ, p. 261.
Murthy, A.S.R. and Naikan, V.N.A. (1996), “Condition monitoring strategy: a risk-based interval
selection”, Int. J. Prod. Res., Vol. 34 No. 1, pp. 285-96.
Pulcini, G. (2001), “Modeling the failure data of a repairable equipment with bathtub type failure
intensity”, Reliability Engineering & System Safety, Vol. 71, pp. 209-18.
Ramakumar, R. (1993), Engineering Reliability, Fundamentals and Applications, Prentice-Hall,
Englewood Cliffs, NJ.
Rao, B.V.A. and Varaprasad, N. (1985), “A theoretical model for increasing the availability of
complex systems”, Maintenance Management International, Vol. 5, pp. 77-81.
Rouhan, A. and Schoefs, F. (2003), “Probabilistic modeling of inspection results for offshore
structures”, Structural Safety, Vol. 25 No. 4, pp. 379-99.
Sarker, R. and Haque, A. (2000), “Optimization of maintenance and spare provisioning policy
using simulation”, Applied Mathematical Modelling, Vol. 24, pp. 751-60.
Wild, R. (1995), Essentials of Production & Operations Management, Redwood Books Ltd,
London.
Xie, M., Tang, Y. and Goh, T.N. (2002), “A modified Weibull extension with bathtub-shaped
failure rate function”, Reliability Engineering & System Safety, Vol. 76, pp. 279-85.
Optimal
inspection
frequency
771